CHAPTER : INTEGRATION Content pge Concept Mp 4. Integrtion of Algeric Functions 4 Eercise A 5 4. The Eqution of Curve from Functions of Grdients. 6 Ee
|
|
|
- Pauline Loreen Newton
- 6 years ago
- Views:
Transcription
1 ADDITIONAL MATHEMATICS FORM 5 MODULE 4 INTEGRATION
2 CHAPTER : INTEGRATION Content pge Concept Mp 4. Integrtion of Algeric Functions 4 Eercise A 5 4. The Eqution of Curve from Functions of Grdients. 6 Eercise B 7 SPM Question 8 9 Assessment Answer
3 n n c ). d = = + d c n + n + Indefinite Integrl ò ) d = + c. n+ n ) ò d = + c. n + ò ò Integrtion of Algeric Functions ò [ ± ] = ò ± ò e) d) The Eqution f ( ) g( ) ofd Curve f ( from ) d g( ) Functions of Grdients ò y = f '( ) d y = f ( ) + c,
4 INTEGRATION. Integrtion is the reverse process of differentition. dy. If y is function of nd = f '( ) then '( ) = +, = constnt. d ò f d y c c dy If = f ( ), then ò f ( ) d = y d 4.. Integrtion of Algeric Functions Indefinite Integrl ò ) d = + c. nd c re constnts n+ n ) ò d = + c. n + c is constnt, n is n integer nd n - ò n+ n n ò c) d = d = + c. n + nd c re constnts n is n ò [ ± ] = ò ± ò d) f ( ) g( ) d f ( ) d g( ) d
5 Find the indefinite integrl for ech of the following. ) 5 d ) d 5 c) d d) ( ) d Alwys rememer to include +c in your nswers of indefinite integrls. Solution : 4 = c 4 ) 5d 5 c ) d c d) 5 ( ) d d d 6 = c = c 6 6 = c = c 5 5 c) d c Find the indefinite integrl for ech of the following. 4 ) d ) d 4 4 Solution : ) d d d = c = c 4 4 ) d = d 4 = 4 d = 4 c 4 = c 4
6 . Find 4 d. [m]. Find d. [m]. Find d. [m] 4. Find d. 4 [m] Integrte with respect to. [m] 6. Find d [m] 7. Find 6 d 6. [m] 8. Integrte with respect to. [m] 5
7 The Eqution of Curve from Functions of Grdients dy If the grdient function of the curve is f '( ), d then the eqution of the curve is y f '( ) d y f ( ) c, c is constnt. Find the eqution of the curve tht hs the grdient function nd psses through the point (, ). Solution The grdient function is. dy d y ( ) d y c The curve psses through the point(, ). Thus, =, y=. () c 6 4 c c 5 Hence, the eqution of curve is y 5 6
8 dy. Given tht 6, epress y in terms of if y = 9 when =. d. Given the grdient function of curve is 4. Find the eqution of the curve if it psses through the point (, 6). dy 48. The grdient function of curve is given y k, where k is constnt. d Given tht the tngent to the curve t the point (-, 4) is prllel to the -is, find the eqution of the curve. 7
9 SPM - Pper :Question () dy Given tht nd y = 6 when =, find y in terms of. [ mrks] d SPM 4- Pper :Question 5() The grdient function of curve which psses through A(, ) is eqution of the curve. 6. Find the [ mrks] 8
10 SPM 5- Pper :Question A curve hs grdient function p 4, where p is constnt. The tngent to the curve t the point (, ) is prllel to the stright line y + 5 =. Find () the vlue of p, [ mrks] () the eqution of the curve. [ mrks] 9
11 . Find the indefinite integrl for ech of the following. 6 d d () 4 () ( c) 6 5 (c) + d (d) 5 d. If dy d 4 4, nd y = when =, find y in terms of.
12 . If dp dv v v, nd p = when v =, find the vlue of p when v =. 4. Find the eqution of the curve with grdient origin., which psses through the 5. Given tht of. d y dy dy 4, nd tht, y = when =. Find d d d nd y in terms
13 EXERCISE A ) c ) ) 4 c 4 4 c 4) 5) 6) c 4 c 7) c 8) c EXERCISE B ) y ) y 4 ) y SPM QUESTIONS ) y 7 ) y ) p, y 4 ASSESSMENT c ( ) ( c) c c ( d) 9 6 c 4 ) ( ) 4 ) y 8 7 ) p 8 4) y 5) y
14 ADDITIONAL MATHEMATICS FORM 5 MODULE 5 INTEGRATION
15 CONTENT CONCEPT MAP INTEGRATION BY SUBSTITUTION DEFINITE INTEGRALS 5 EXERCISE A 6 EXERCISE B 7 ASSESSMENT 8 SPM QUESTIOS 9 ANSWERS 4
16 CONCEPT MAP INTEGRATION BY SUBSTITUTION u n d du where u = +, nd re constnts, n is n integer nd n - n OR n n d c, n where,, nd c re constnts, n is integer nd n - DEFINITE INTEGRALS d If g ( ) f ( ) then d () f( ) d g( ) g( ) g( ) ( ) f( ) d f( ) d c c ( c) f( ) d f( ) d f( ) d 5
17 INTEGRATION BY SUBSTITUTION u n d du where u = +, nd re constnts, n is n integer nd n - n O R n n d c, n where,, nd c re constnts, n is integer nd n - Find the indefinite integrl for ech of the following. () d 7 () 4( 5) d (c) (5 ) d SOLUTION () d Let u = + du du d d du ( ) d u u = du u = c ( ) Sustitute + nd sustitute d with du d = OR ( ) (4) 4 d c ( ) = 8 4 c 4 u = + c 8 ( ) = + c 8 Sustitute u = + 6
18 () 7 4( 5) d Let u 5 du du d d 7 7 4u 4( 5) d du 8 4u = c (8) OR 8 u = c 6 (u 5) = 6 8 c (c) (5 ) d (5 ) d Let u 5 du du 5 d d 5 u (5 ) d du 5 u = c 5( ) u = c 5 = 5u = c 5(5 ) 4( 5) (8) 8 7 d c 4( 5) 8 ( 5) = c 6 DEFINITE INTEGRALS d If g ( ) f ( ) then d () f ( ) d g( ) g( ) g( ) ( ) f ( ) d f ( ) d c c ( c) f ( ) d f ( ) d f ( ) d 7
19 Evlute ech of the following ( )( ) () d 4 () ( ) d SOLUTION () ( )( ) 9 c 4 d 4 9 = d 4 4 = ( 9 ) d 4 = 9 = = = ( ) 8 = 8 = 8 () ( ) d ( ) d = ( ) ( ) = () d = ( ) = [() ] [() ] = 6 = 8
20 EXERCISE A INTEGRATE THE FOLLOWING USING SUBSTITUTION METHOD. () ( ) d () d () 5 d 4 (4) 5 d (5) 5 4 y dy 4 (6) 5 u du 5 9
21 EXERCISE B. Evlute 8 ( 4) d Answer : Evlute ( 5) d 8 Answer: Integrte 5 with respect to Answer: 5 c Evlute d 4 Answer : 5. Evlute Answer: d 6.Given tht of f ( ) d 5 Answer :7 5 f ( ) d, find the vlue
22 ASSESSMENT. Given tht f ( ) d 7. Find f ( ) d () () 5 f ( ) d the vlue of k if k f ( ) d 8 nd.() 5( v) 4 () dv d Answer : () k = () 48. Show tht Hence, find the vlue of Answer : d d 6 d.. 4. Given tht 4 g( ) d 5. Find 4 () 4 4 () f ( ) ( ) f ( ) d g( ) d Answer: () 5 () 4 g d 4 f ( ) d nd
23 SPM QUESTIONS SPM PAPER, QUESTION 7.Given tht 5 n d k 4 c, find the vlue of k nd n [ mrks] 5 Answer: k = n = - SPM 4 PAPER, QUESTION k. Given tht d 6, where k > -, find the vlue of k. [4 mrks] Answer: k = 5 SPM 5 PAPER, QUESTION. Given tht 6 f ( ) d 7 nd 6 ( f ( ) k) d, find the vlue of k. Answer: k = 4
24 ANSWERS EXERCISE A. ( + ) 4 + c. 6 ( +5) c. c c y c u c 6 EXERCISE B c ASSESSMENT. () k = () 48. () 9( v) -5 +c () ( 5 ) 4 c SPM QUESTIONS 5. k = n = -. k = 5. k = () 5 () 4
25 ADDITIONAL MATHEMATICS MODULE 6 INTEGRATION 4
26 CHAPTER : INTEGRATION Content pge Concept Mp 9. Integrtion s Summtion of Ares Eercise A Integrtion s Summtion of Volumes 7 8 Eercise B 9 SPM Question 4 Answer 5 5
27 ) The re under curve which enclosed y -is, = nd = is y d ) The re under curve which enclosed y y-is, y = nd y = is dy c) The re enclosed y curve nd stright line f ( ) g( ) d ) The volume generted when curve is rotted through 6º out the -is is V y d ) The volume generted when curve is rotted through 6º out the y-is is V y dy 6
28 . INTEGRATION. Integrtion s Summtion of Are y = f() y y = f() The re under curve which enclosed y = nd = is yd Note : The re is preceded y negtive sign if the region lies elow the is. The re under curve which is enclosed y y = nd y = is dy Note : The re is preceded y negtive sign if the region is to the left of the y is. The re enclosed y curve nd stright line y y = g () y = f () The re of the shded region = f ( ) d g( ) = f ( ) ( ) g d 7
29 . Find the re of the shded region in the digrm. y y =. Find the re of the shded region in the digrm. y y =
30 . Find the re of the shded region 4. Find the re of the shded region in the digrm. y y y = y = = y - - 9
31 5. Find the re of the shded region in the digrm y 6. y y = ( ) = y - y = k - Given tht the re of the shded region in the digrm ove is 8 units. Find the vlue of k.
32 . Integrtion s Summtion of Volumes y y=f() The volume generted when curve is rotted through 6º out the -is is V y d y y=f() The volume generted when curve is rotted through 6º out the y-is is V y dy
33 y y=(+) Find the volume generted when the shded region is rotted through 6º out the -is. Answer : = Volume generted y d d 4 ( ) d () units. Answer : Given y 6 y y 6 sustitute into y 6 Then, y 6 y 6 Volume generted The figure shows the shded region tht is enclosed y the curve y 6, the -is nd the y-is. Clculte the volume generted when the shded region is revolved through 6º out y-is. 6 dy 6 6 y d y 6y 6 6(6) 8 units. 6
34 . y y = ( ) The ove figure shows the shded region tht is enclosed y the curve y = ( ) nd -is. Clculte the volume generted when the shded region is revolved through 6º out the y-is. [4 mrks]
35 . y R (, 4) Q (, 4) P (, ) y² = 4 ( + ) = The figure shows the curve y ( ). Clculte the volume generted when the shded region is revolved through 6º out the -is. 4
36 . y R (, 4) y = k The ove figure shows prt of the curve y nd the stright line = k. If the volume generted when the shded region is revolved through 6º out the -is is units, find the vlue of k. 5
37 y SPM - Pper :Question 9 () Digrm shows curve y which intersects the stright line y t point A. y y y Digrm Clculte the volume generted when the shded region is involved 6º out the y-is. [6 mrks] 6
38 SPM 4- Pper :Question Digrm 5 shows prt of the curve y y which psses through A(, ). A(,) Digrm 5 y ) Find the eqution of the tngent to the curve t the point A. [4 mrks] ) A region is ounded y the curve, the -is nd the stright lines = nd =. i) Find the re of the region. ii) The region is revolved through 6º out the -is. Find the volume generted, in terms of. [6 mrks] 7
39 SPM 5- Pper :Question In Digrm 4, the stright line PQ is norml to the curve stright line AR is prllel to the y-is. y y y t A(, ). The A(, ) Q(k, ) R Digrm 4 Find () the vlue of k, [ mrks] () the re of the shded region, [4 mrks] (c) the volume generted, in terms of, when the region ounded y the curve, the y-is nd the stright line y = is revolved through 6º out y-is. [ mrks] 8
40 EXERCISE A. units 5. units 6. units 4. 4 units 5. units 6. k 4 EXERCISE B. unit 5. 6 unit 5. k SPM QUESTIONS SPM 5 Volume Generted units 5 SPM 4 i) Are units 5 ii) 49 Volume Generted units 5 SPM 5 ) k 8 ) Are units c) Volume Generted 4 units 9
Calculus 2: Integration. Differentiation. Integration
Clculus 2: Integrtion The reverse process to differentition is known s integrtion. Differentition f() f () Integrtion As it is the opposite of finding the derivtive, the function obtined b integrtion is
Thomas Whitham Sixth Form
Thoms Whithm Sith Form Pure Mthemtics Unit C Alger Trigonometry Geometry Clculus Vectors Trigonometry Compound ngle formule sin sin cos cos Pge A B sin Acos B cos Asin B A B sin Acos B cos Asin B A B cos
R(3, 8) P( 3, 0) Q( 2, 2) S(5, 3) Q(2, 32) P(0, 8) Higher Mathematics Objective Test Practice Book. 1 The diagram shows a sketch of part of
Higher Mthemtics Ojective Test Prctice ook The digrm shows sketch of prt of the grph of f ( ). The digrm shows sketch of the cuic f ( ). R(, 8) f ( ) f ( ) P(, ) Q(, ) S(, ) Wht re the domin nd rnge of
Mathematics. Area under Curve.
Mthemtics Are under Curve www.testprepkrt.com Tle of Content 1. Introduction.. Procedure of Curve Sketching. 3. Sketching of Some common Curves. 4. Are of Bounded Regions. 5. Sign convention for finding
FINALTERM EXAMINATION 9 (Session - ) Clculus & Anlyticl Geometry-I Question No: ( Mrs: ) - Plese choose one f ( x) x According to Power-Rule of differentition, if d [ x n ] n x n n x n n x + ( n ) x n+
A LEVEL TOPIC REVIEW. factor and remainder theorems
A LEVEL TOPIC REVIEW unit C fctor nd reminder theorems. Use the Fctor Theorem to show tht: ) ( ) is fctor of +. ( mrks) ( + ) is fctor of ( ) is fctor of + 7+. ( mrks) +. ( mrks). Use lgebric division
FORM FIVE ADDITIONAL MATHEMATIC NOTE. ar 3 = (1) ar 5 = = (2) (2) (1) a = T 8 = 81
FORM FIVE ADDITIONAL MATHEMATIC NOTE CHAPTER : PROGRESSION Arithmetic Progression T n = + (n ) d S n = n [ + (n )d] = n [ + Tn ] S = T = T = S S Emple : The th term of n A.P. is 86 nd the sum of the first
Improper Integrals. Introduction. Type 1: Improper Integrals on Infinite Intervals. When we defined the definite integral.
Improper Integrls Introduction When we defined the definite integrl f d we ssumed tht f ws continuous on [, ] where [, ] ws finite, closed intervl There re t lest two wys this definition cn fil to e stisfied:
KEY CONCEPTS. satisfies the differential equation da. = 0. Note : If F (x) is any integral of f (x) then, x a
KEY CONCEPTS THINGS TO REMEMBER :. The re ounded y the curve y = f(), the -is nd the ordintes t = & = is given y, A = f () d = y d.. If the re is elow the is then A is negtive. The convention is to consider
( ) Straight line graphs, Mixed Exercise 5. 2 b The equation of the line is: 1 a Gradient m= 5. The equation of the line is: y y = m x x = 12.
Stright line grphs, Mied Eercise Grdient m ( y ),,, The eqution of the line is: y m( ) ( ) + y + Sustitute (k, ) into y + k + k k Multiply ech side y : k k The grdient of AB is: y y So: ( k ) 8 k k 8 k
y = f(x) This means that there must be a point, c, where the Figure 1
Clculus Investigtion A Men Slope TEACHER S Prt 1: Understnding the Men Vlue Theorem The Men Vlue Theorem for differentition sttes tht if f() is defined nd continuous over the intervl [, ], nd differentile
Calculus Module C21. Areas by Integration. Copyright This publication The Northern Alberta Institute of Technology All Rights Reserved.
Clculus Module C Ares Integrtion Copright This puliction The Northern Alert Institute of Technolog 7. All Rights Reserved. LAST REVISED Mrch, 9 Introduction to Ares Integrtion Sttement of Prerequisite
Year 12 Mathematics Extension 2 HSC Trial Examination 2014
Yer Mthemtics Etension HSC Tril Emintion 04 Generl Instructions. Reding time 5 minutes Working time hours Write using blck or blue pen. Blck pen is preferred. Bord-pproved clcultors my be used A tble of
MATHEMATICS PAPER IIB COORDINATE GEOMETRY AND CALCULUS. Note: This question paper consists of three sections A,B and C. SECTION A
MATHEMATICS PAPER IIB COORDINATE GEOMETRY AND CALCULUS. TIME : 3hrs M. Mrks.75 Note: This question pper consists of three sections A,B nd C. SECTION A VERY SHORT ANSWER TYPE QUESTIONS. X = ) Find the eqution
Math 190 Chapter 5 Lecture Notes. Professor Miguel Ornelas
Mth 19 Chpter 5 Lecture Notes Professor Miguel Ornels 1 M. Ornels Mth 19 Lecture Notes Section 5.1 Section 5.1 Ares nd Distnce Definition The re A of the region S tht lies under the grph of the continuous
Polynomials and Division Theory
Higher Checklist (Unit ) Higher Checklist (Unit ) Polynomils nd Division Theory Skill Achieved? Know tht polynomil (expression) is of the form: n x + n x n + n x n + + n x + x + 0 where the i R re the
Mathematics Extension 2
00 HIGHER SCHOOL CERTIFICATE EXAMINATION Mthemtics Etension Generl Instructions Reding time 5 minutes Working time hours Write using blck or blue pen Bord-pproved clcultors my be used A tble of stndrd
2 b. , a. area is S= 2π xds. Again, understand where these formulas came from (pages ).
AP Clculus BC Review Chpter 8 Prt nd Chpter 9 Things to Know nd Be Ale to Do Know everything from the first prt of Chpter 8 Given n integrnd figure out how to ntidifferentite it using ny of the following
Worksheet A EXPONENTIALS AND LOGARITHMS PMT. 1 Express each of the following in the form log a b = c. a 10 3 = 1000 b 3 4 = 81 c 256 = 2 8 d 7 0 = 1
C Worksheet A Epress ech of the following in the form log = c. 0 = 000 4 = 8 c 56 = 8 d 7 0 = e = f 5 = g 7 9 = 9 h 6 = 6 Epress ech of the following using inde nottion. log 5 5 = log 6 = 4 c 5 = log 0
Session Trimester 2. Module Code: MATH08001 MATHEMATICS FOR DESIGN
School of Science & Sport Pisley Cmpus Session 05-6 Trimester Module Code: MATH0800 MATHEMATICS FOR DESIGN Dte: 0 th My 06 Time: 0.00.00 Instructions to Cndidtes:. Answer ALL questions in Section A. Section
Math 259 Winter Solutions to Homework #9
Mth 59 Winter 9 Solutions to Homework #9 Prolems from Pges 658-659 (Section.8). Given f(, y, z) = + y + z nd the constrint g(, y, z) = + y + z =, the three equtions tht we get y setting up the Lgrnge multiplier
( x )( x) dx. Year 12 Extension 2 Term Question 1 (15 Marks) (a) Sketch the curve (x + 1)(y 2) = 1 2
Yer Etension Term 7 Question (5 Mrks) Mrks () Sketch the curve ( + )(y ) (b) Write the function in prt () in the form y f(). Hence, or otherwise, sketch the curve (i) y f( ) (ii) y f () (c) Evlute (i)
Math RE - Calculus II Area Page 1 of 12
Mth --RE - Clculus II re Pge of re nd the Riemnn Sum Let f) be continuous function nd = f) f) > on closed intervl,b] s shown on the grph. The Riemnn Sum theor shows tht the re of R the region R hs re=
SULIT /2 3472/2 Matematik Tambahan Kertas 2 2 ½ jam 2009 SEKOLAH-SEKOLAH MENENGAH ZON A KUCHING
SULIT 1 347/ 347/ Mtemtik Tmbhn Kerts ½ jm 009 SEKOLAH-SEKOLAH MENENGAH ZON A KUCHING PEPERIKSAAN PERCUBAAN SIJIL PELAJARAN MALAYSIA 009 MATEMATIK TAMBAHAN Kerts Du jm tig puluh minit JANGAN BUKA KERTAS
S56 (5.3) Vectors.notebook January 29, 2016
Dily Prctice 15.1.16 Q1. The roots of the eqution (x 1)(x + k) = 4 re equl. Find the vlues of k. Q2. Find the rte of chnge of 剹 x when x = 1 / 8 Tody we will e lerning out vectors. Q3. Find the eqution
Evaluating Definite Integrals. There are a few properties that you should remember in order to assist you in evaluating definite integrals.
Evluting Definite Integrls There re few properties tht you should rememer in order to ssist you in evluting definite integrls. f x dx= ; where k is ny rel constnt k f x dx= k f x dx ± = ± f x g x dx f
Indefinite Integral. Chapter Integration - reverse of differentiation
Chpter Indefinite Integrl Most of the mthemticl opertions hve inverse opertions. The inverse opertion of differentition is clled integrtion. For exmple, describing process t the given moment knowing the
AB Calculus Review Sheet
AB Clculus Review Sheet Legend: A Preclculus, B Limits, C Differentil Clculus, D Applictions of Differentil Clculus, E Integrl Clculus, F Applictions of Integrl Clculus, G Prticle Motion nd Rtes This is
Mathematics Extension Two
Student Number 04 HSC TRIAL EXAMINATION Mthemtics Etension Two Generl Instructions Reding time 5 minutes Working time - hours Write using blck or blue pen Bord-pproved clcultors my be used Write your Student
( ) where f ( x ) is a. AB Calculus Exam Review Sheet. A. Precalculus Type problems. Find the zeros of f ( x).
AB Clculus Exm Review Sheet A. Preclculus Type prolems A1 Find the zeros of f ( x). This is wht you think of doing A2 A3 Find the intersection of f ( x) nd g( x). Show tht f ( x) is even. A4 Show tht f
( ) as a fraction. Determine location of the highest
AB Clculus Exm Review Sheet - Solutions A. Preclculus Type prolems A1 A2 A3 A4 A5 A6 A7 This is wht you think of doing Find the zeros of f ( x). Set function equl to 0. Fctor or use qudrtic eqution if
14.4. Lengths of curves and surfaces of revolution. Introduction. Prerequisites. Learning Outcomes
Lengths of curves nd surfces of revolution 4.4 Introduction Integrtion cn be used to find the length of curve nd the re of the surfce generted when curve is rotted round n xis. In this section we stte
Higher Checklist (Unit 3) Higher Checklist (Unit 3) Vectors
Vectors Skill Achieved? Know tht sclr is quntity tht hs only size (no direction) Identify rel-life exmples of sclrs such s, temperture, mss, distnce, time, speed, energy nd electric chrge Know tht vector
Section - 2 MORE PROPERTIES
LOCUS Section - MORE PROPERTES n section -, we delt with some sic properties tht definite integrls stisf. This section continues with the development of some more properties tht re not so trivil, nd, when
First midterm topics Second midterm topics End of quarter topics. Math 3B Review. Steve. 18 March 2009
Mth 3B Review Steve 18 Mrch 2009 About the finl Fridy Mrch 20, 3pm-6pm, Lkretz 110 No notes, no book, no clcultor Ten questions Five review questions (Chpters 6,7,8) Five new questions (Chpters 9,10) No
( ) where f ( x ) is a. AB/BC Calculus Exam Review Sheet. A. Precalculus Type problems. Find the zeros of f ( x).
AB/ Clculus Exm Review Sheet A. Preclculus Type prolems A1 Find the zeros of f ( x). This is wht you think of doing A2 Find the intersection of f ( x) nd g( x). A3 Show tht f ( x) is even. A4 Show tht
Year 12 Trial Examination Mathematics Extension 1. Question One 12 marks (Start on a new page) Marks
THGS Mthemtics etension Tril 00 Yer Tril Emintion Mthemtics Etension Question One mrks (Strt on new pge) Mrks ) If P is the point (-, 5) nd Q is the point (, -), find the co-ordintes of the point R which
Believethatyoucandoitandyouar. Mathematics. ngascannotdoonlynotyetbelieve thatyoucandoitandyouarehalfw. Algebra
Believethtoucndoitndour ehlfwtherethereisnosuchthi Mthemtics ngscnnotdoonlnotetbelieve thtoucndoitndourehlfw Alger therethereisnosuchthingsc nnotdoonlnotetbelievethto Stge 6 ucndoitndourehlfwther S Cooper
x ) dx dx x sec x over the interval (, ).
Curve on 6 For -, () Evlute the integrl, n (b) check your nswer by ifferentiting. ( ). ( ). ( ).. 6. sin cos 7. sec csccot 8. sec (sec tn ) 9. sin csc. Evlute the integrl sin by multiplying the numertor
7.1 Integral as Net Change and 7.2 Areas in the Plane Calculus
7.1 Integrl s Net Chnge nd 7. Ares in the Plne Clculus 7.1 INTEGRAL AS NET CHANGE Notecrds from 7.1: Displcement vs Totl Distnce, Integrl s Net Chnge We hve lredy seen how the position of n oject cn e
sec x over the interval (, ). x ) dx dx x 14. Use a graphing utility to generate some representative integral curves of the function Curve on 5
Curve on Clcultor eperience Fin n ownlo (or type in) progrm on your clcultor tht will fin the re uner curve using given number of rectngles. Mke sure tht the progrm fins LRAM, RRAM, n MRAM. (You nee to
HIGHER SCHOOL CERTIFICATE EXAMINATION MATHEMATICS 3 UNIT (ADDITIONAL) AND 3/4 UNIT (COMMON) Time allowed Two hours (Plus 5 minutes reading time)
HIGHER SCHOOL CERTIFICATE EXAMINATION 998 MATHEMATICS 3 UNIT (ADDITIONAL) AND 3/4 UNIT (COMMON) Time llowed Two hours (Plus 5 minutes reding time) DIRECTIONS TO CANDIDATES Attempt ALL questions ALL questions
( β ) touches the x-axis if = 1
Generl Certificte of Eduction (dv. Level) Emintion, ugust Comined Mthemtics I - Prt B Model nswers. () Let f k k, where k is rel constnt. i. Epress f in the form( ) Find the turning point of f without
Section 4: Integration ECO4112F 2011
Reding: Ching Chpter Section : Integrtion ECOF Note: These notes do not fully cover the mteril in Ching, ut re ment to supplement your reding in Ching. Thus fr the optimistion you hve covered hs een sttic
2. VECTORS AND MATRICES IN 3 DIMENSIONS
2 VECTORS AND MATRICES IN 3 DIMENSIONS 21 Extending the Theory of 2-dimensionl Vectors x A point in 3-dimensionl spce cn e represented y column vector of the form y z z-xis y-xis z x y x-xis Most of the
June 2011 Further Pure Mathematics FP Mark Scheme
. June 0 Further Pure Mthemtics FP 6669 Mrk dy 6x dx = nd so surfce re = π x ( + (6 x ) dx B 4 = 4 π ( 6 x ) + 6 4 4π D 860.06 = 806 (to sf) 6 Use limits nd 0 to give [ ] B Both bits CAO but condone lck
Ch AP Problems
Ch. 7.-7. AP Prolems. Willy nd his friends decided to rce ech other one fternoon. Willy volunteered to rce first. His position is descried y the function f(t). Joe, his friend from school, rced ginst him,
Mathematics Extension 2
S Y D N E Y B O Y S H I G H S C H O O L M O O R E P A R K, S U R R Y H I L L S 005 HIGHER SCHOOL CERTIFICATE TRIAL PAPER Mthemtics Extension Generl Instructions Totl Mrks 0 Reding Time 5 Minutes Attempt
HIGHER SCHOOL CERTIFICATE EXAMINATION MATHEMATICS 4 UNIT (ADDITIONAL) Time allowed Three hours (Plus 5 minutes reading time)
HIGHER SCHOOL CERTIFICATE EXAMINATION 999 MATHEMATICS UNIT (ADDITIONAL) Time llowed Three hours (Plus 5 minutes reding time) DIRECTIONS TO CANDIDATES Attempt ALL questions ALL questions re of equl vlue
Review Problem for Midterm #1
Review Problem for Midterm # Midterm I: - :5.m Fridy (Sep. ), Topics: 5.8 nd 6.-6. Office Hours before the midterm I: W - pm (Sep 8), Th -5 pm (Sep 9) t UH 8B Solution to quiz cn be found t http://mth.utoledo.edu/
UNIVERSITY OF BOLTON SCHOOL OF ENGINEERING B.ENG (HONS) ELECTRICAL AND ELECTRONIC ENGINEERING EXAMINATION SEMESTER /2018
ENG005 B.ENG (HONS) ELECTRICAL AND ELECTRONIC ENGINEERING EXAMINATION SEMESTER 1-017/018 MODULE NO: EEE4001 Dte: 19Jnury 018 Time:.00 4.00 INSTRUCTIONS TO CANDIDATES: There re SIX questions. Answer ANY
ONLINE PAGE PROOFS. Anti-differentiation and introduction to integral calculus
Anti-differentition nd introduction to integrl clculus. Kick off with CAS. Anti-derivtives. Anti-derivtive functions nd grphs. Applictions of nti-differentition.5 The definite integrl.6 Review . Kick off
Review Exercises for Chapter 4
_R.qd // : PM Pge CHAPTER Integrtion Review Eercises for Chpter In Eercises nd, use the grph of to sketch grph of f. To print n enlrged cop of the grph, go to the wesite www.mthgrphs.com... In Eercises
03 Qudrtic Functions Completing the squre: Generl Form f ( x) x + x + c f ( x) ( x + p) + q where,, nd c re constnts nd 0. (i) (ii) (iii) (iv) *Note t
A-PDF Wtermrk DEMO: Purchse from www.a-pdf.com to remove the wtermrk Add Mths Formule List: Form 4 (Updte 8/9/08) 0 Functions Asolute Vlue Function Inverse Function If f ( x ), if f ( x ) 0 f ( x) y f
Mathematics Extension 2
00 HIGHER SCHOOL CERTIFICATE EXAMINATION Mthemtics Extension Generl Instructions Reding time 5 minutes Working time hours Write using blck or blue pen Bord-pproved clcultors m be used A tble of stndrd
Mathematics Higher Block 3 Practice Assessment A
Mthemtics Higher Block 3 Prctice Assessment A Red crefully 1. Clcultors my be used. 2. Full credit will be given only where the solution contins pproprite working. 3. Answers obtined from reding from scle
Chapter 6 Techniques of Integration
MA Techniques of Integrtion Asst.Prof.Dr.Suprnee Liswdi Chpter 6 Techniques of Integrtion Recll: Some importnt integrls tht we hve lernt so fr. Tle of Integrls n+ n d = + C n + e d = e + C ( n ) d = ln
x = b a n x 2 e x dx. cdx = c(b a), where c is any constant. a b
CHAPTER 5. INTEGRALS 61 where nd x = b n x i = 1 (x i 1 + x i ) = midpoint of [x i 1, x i ]. Problem 168 (Exercise 1, pge 377). Use the Midpoint Rule with the n = 4 to pproximte 5 1 x e x dx. Some quick
Coimisiún na Scrúduithe Stáit State Examinations Commission
M 30 Coimisiún n Scrúduithe Stáit Stte Exmintions Commission LEAVING CERTIFICATE EXAMINATION, 005 MATHEMATICS HIGHER LEVEL PAPER ( 300 mrks ) MONDAY, 3 JUNE MORNING, 9:30 to :00 Attempt FIVE questions
Bob Brown Math 251 Calculus 1 Chapter 5, Section 4 1 CCBC Dundalk
Bo Brown Mth Clculus Chpter, Section CCBC Dundlk The Fundmentl Theorem of Clculus Informlly, the Fundmentl Theorem of Clculus (FTC) sttes tht differentition nd definite integrtion re inverse opertions
CONIC SECTIONS. Chapter 11
CONIC SECTIONS Chpter. Overview.. Sections of cone Let l e fied verticl line nd m e nother line intersecting it t fied point V nd inclined to it t n ngle α (Fig..). Fig.. Suppose we rotte the line m round
RAM RAJYA MORE, SIWAN. XI th, XII th, TARGET IIT-JEE (MAIN + ADVANCE) & COMPATETIVE EXAM FOR XII (PQRS) INDEFINITE INTERATION & Their Properties
M.Sc. (Mths), B.Ed, M.Phil (Mths) MATHEMATICS Mob. : 947084408 9546359990 M.Sc. (Mths), B.Ed, M.Phil (Mths) RAM RAJYA MORE, SIWAN XI th, XII th, TARGET IIT-JEE (MAIN + ADVANCE) & COMPATETIVE EXAM FOR XII
6.2 CONCEPTS FOR ADVANCED MATHEMATICS, C2 (4752) AS
6. CONCEPTS FOR ADVANCED MATHEMATICS, C (475) AS Objectives To introduce students to number of topics which re fundmentl to the dvnced study of mthemtics. Assessment Emintion (7 mrks) 1 hour 30 minutes.
Chapter 1: Logarithmic functions and indices
Chpter : Logrithmic functions nd indices. You cn simplify epressions y using rules of indices m n m n m n m n ( m ) n mn m m m m n m m n Emple Simplify these epressions: 5 r r c 4 4 d 6 5 e ( ) f ( ) 4
Unit #10 De+inite Integration & The Fundamental Theorem Of Calculus
Unit # De+inite Integrtion & The Fundmentl Theorem Of Clculus. Find the re of the shded region ove nd explin the mening of your nswer. (squres re y units) ) The grph to the right is f(x) = -x + 8x )Use
LINEAR ALGEBRA APPLIED
5.5 Applictions of Inner Product Spces 5.5 Applictions of Inner Product Spces 7 Find the cross product of two vectors in R. Find the liner or qudrtic lest squres pproimtion of function. Find the nth-order
MATH 144: Business Calculus Final Review
MATH 144: Business Clculus Finl Review 1 Skills 1. Clculte severl limits. 2. Find verticl nd horizontl symptotes for given rtionl function. 3. Clculte derivtive by definition. 4. Clculte severl derivtives
13.3. The Area Bounded by a Curve. Introduction. Prerequisites. Learning Outcomes
The Are Bounded b Curve 3.3 Introduction One of the importnt pplictions of integrtion is to find the re bounded b curve. Often such n re cn hve phsicl significnce like the work done b motor, or the distnce
Mat 210 Updated on April 28, 2013
Mt Brief Clculus Mt Updted on April 8, Alger: m n / / m n m n / mn n m n m n n ( ) ( )( ) n terms n n n n n n ( )( ) Common denomintor: ( ) ( )( ) ( )( ) ( )( ) ( )( ) Prctice prolems: Simplify using common
Calculus AB. For a function f(x), the derivative would be f '(
lculus AB Derivtive Formuls Derivtive Nottion: For function f(), the derivtive would e f '( ) Leiniz's Nottion: For the derivtive of y in terms of, we write d For the second derivtive using Leiniz's Nottion:
MTH 4-16a Trigonometry
MTH 4-16 Trigonometry Level 4 [UNIT 5 REVISION SECTION ] I cn identify the opposite, djcent nd hypotenuse sides on right-ngled tringle. Identify the opposite, djcent nd hypotenuse in the following right-ngled
Edexcel GCE Core Mathematics (C2) Required Knowledge Information Sheet. Daniel Hammocks
Edexcel GCE Core Mthemtics (C) Required Knowledge Informtion Sheet C Formule Given in Mthemticl Formule nd Sttisticl Tles Booklet Cosine Rule o = + c c cosine (A) Binomil Series o ( + ) n = n + n 1 n 1
Mathematics Extension 1
04 Bored of Studies Tril Emintions Mthemtics Etension Written by Crrotsticks & Trebl. Generl Instructions Totl Mrks 70 Reding time 5 minutes. Working time hours. Write using blck or blue pen. Blck pen
Section 6: Area, Volume, and Average Value
Chpter The Integrl Applied Clculus Section 6: Are, Volume, nd Averge Vlue Are We hve lredy used integrls to find the re etween the grph of function nd the horizontl xis. Integrls cn lso e used to find
MEP Practice Book ES3. 1. Calculate the size of the angles marked with a letter in each diagram. None to scale
ME rctice ook ES3 3 ngle Geometr 3.3 ngle Geometr 1. lculte the size of the ngles mrked with letter in ech digrm. None to scle () 70 () 20 54 65 25 c 36 (d) (e) (f) 56 62 d e 60 40 70 70 f 30 g (g) (h)
Math 1102: Calculus I (Math/Sci majors) MWF 3pm, Fulton Hall 230 Homework 2 solutions
Mth 1102: Clculus I (Mth/Sci mjors) MWF 3pm, Fulton Hll 230 Homework 2 solutions Plese write netly, nd show ll work. Cution: An nswer with no work is wrong! Do the following problems from Chpter III: 6,
Topics Covered AP Calculus AB
Topics Covered AP Clculus AB ) Elementry Functions ) Properties of Functions i) A function f is defined s set of ll ordered pirs (, y), such tht for ech element, there corresponds ectly one element y.
Chapter 9 Definite Integrals
Chpter 9 Definite Integrls In the previous chpter we found how to tke n ntiderivtive nd investigted the indefinite integrl. In this chpter the connection etween ntiderivtives nd definite integrls is estlished
HOMEWORK SOLUTIONS MATH 1910 Sections 7.9, 8.1 Fall 2016
HOMEWORK SOLUTIONS MATH 9 Sections 7.9, 8. Fll 6 Problem 7.9.33 Show tht for ny constnts M,, nd, the function yt) = )) t ) M + tnh stisfies the logistic eqution: y SOLUTION. Let Then nd Finlly, y = y M
Section 7.1 Area of a Region Between Two Curves
Section 7.1 Are of Region Between Two Curves White Bord Chllenge The circle elow is inscried into squre: Clcultor 0 cm Wht is the shded re? 400 100 85.841cm White Bord Chllenge Find the re of the region
Bridging the gap: GCSE AS Level
Bridging the gp: GCSE AS Level CONTENTS Chpter Removing rckets pge Chpter Liner equtions Chpter Simultneous equtions 8 Chpter Fctors 0 Chpter Chnge the suject of the formul Chpter 6 Solving qudrtic equtions
SAINT IGNATIUS COLLEGE
SAINT IGNATIUS COLLEGE Directions to Students Tril Higher School Certificte 0 MATHEMATICS Reding Time : 5 minutes Totl Mrks 00 Working Time : hours Write using blue or blck pen. (sketches in pencil). This
Multiplying integers EXERCISE 2B INDIVIDUAL PATHWAYS. -6 ì 4 = -6 ì 0 = 4 ì 0 = -6 ì 3 = -5 ì -3 = 4 ì 3 = 4 ì 2 = 4 ì 1 = -5 ì -2 = -6 ì 2 = -6 ì 1 =
EXERCISE B INDIVIDUAL PATHWAYS Activity -B- Integer multipliction doc-69 Activity -B- More integer multipliction doc-698 Activity -B- Advnced integer multipliction doc-699 Multiplying integers FLUENCY
8 SETS, VECTORS AND FUNCTIONS
1 8 SETS, VECTORS D FUCTIOS TSK 8.1 1. Given tht e ¼ {1,, 3,..., 0}, list the memers of the following susets: ¼ {prime numers} ¼ {odd numers} c \ d [ e \ f \. The figures in the digrm indicte the numer
MAT137 Calculus! Lecture 28
officil wesite http://uoft.me/mat137 MAT137 Clculus! Lecture 28 Tody: Antiderivtives Fundmentl Theorem of Clculus Net: More FTC (review v. 8.5-8.7) 5.7 Sustitution (v. 9.1-9.4) Properties of the Definite
A B= ( ) because from A to B is 3 right, 2 down.
8. Vectors nd vector nottion Questions re trgeted t the grdes indicted Remember: mgnitude mens size. The vector ( ) mens move left nd up. On Resource sheet 8. drw ccurtely nd lbel the following vectors.
Test 3 Review. Jiwen He. I will replace your lowest test score with the percentage grade from the final exam (provided it is higher).
Test 3 Review Jiwen He Test 3 Test 3: Dec. 4-6 in CASA Mteril - Through 6.3. No Homework (Thnksgiving) No homework this week! Hve GREAT Thnksgiving! Finl Exm Finl Exm: Dec. 14-17 in CASA You Might Be Interested
13.4. Integration by Parts. Introduction. Prerequisites. Learning Outcomes
Integrtion by Prts 13.4 Introduction Integrtion by Prts is technique for integrting products of functions. In this Section you will lern to recognise when it is pproprite to use the technique nd hve the
In Mathematics for Construction, we learnt that
III DOUBLE INTEGATION THE ANTIDEIVATIVE OF FUNCTIONS OF VAIABLES In Mthemtics or Construction, we lernt tht the indeinite integrl is the ntiderivtive o ( d ( Double Integrtion Pge Hence d d ( d ( The ntiderivtive
Calculus - Activity 1 Rate of change of a function at a point.
Nme: Clss: p 77 Mths Helper Plus Resource Set. Copright 00 Bruce A. Vughn, Techers Choice Softwre Clculus - Activit Rte of chnge of function t point. ) Strt Mths Helper Plus, then lod the file: Clculus
(b) Let S 1 : f(x, y, z) = (x a) 2 + (y b) 2 + (z c) 2 = 1, this is a level set in 3D, hence
Problem ( points) Find the vector eqution of the line tht joins points on the two lines L : r ( + t) i t j ( + t) k L : r t i + (t ) j ( + t) k nd is perpendiculr to both those lines. Find the set of ll
Math 113 Exam 2 Practice
Mth Em Prctice Februry, 8 Em will cover sections 6.5, 7.-7.5 nd 7.8. This sheet hs three sections. The first section will remind you bout techniques nd formuls tht you should know. The second gives number
Further applications of integration UNCORRECTED PAGE PROOFS
. Kick off with CAS. Integrtion recognition. Solids of revolution. Volumes Further pplictions of integrtion. Arc length, numericl integrtion nd grphs of ntiderivtives.6 Wter flow.7 Review . Kick off with
KOÇ UNIVERSITY MATH 106 FINAL EXAM JANUARY 6, 2013
KOÇ UNIVERSITY MATH 6 FINAL EXAM JANUARY 6, 23 Durtion of Exm: 2 minutes INSTRUCTIONS: No clcultors nd no cell phones my be used on the test. No questions, nd tlking llowed. You must lwys explin your nswers
The area under the graph of f and above the x-axis between a and b is denoted by. f(x) dx. π O
1 Section 5. The Definite Integrl Suppose tht function f is continuous nd positive over n intervl [, ]. y = f(x) x The re under the grph of f nd ove the x-xis etween nd is denoted y f(x) dx nd clled the
PROPERTIES OF AREAS In general, and for an irregular shape, the definition of the centroid at position ( x, y) is given by
PROPERTES OF RES Centroid The concept of the centroid is prol lred fmilir to ou For plne shpe with n ovious geometric centre, (rectngle, circle) the centroid is t the centre f n re hs n is of smmetr, the
FP3 past questions - conics
Hperolic functions cosh sinh = sinh = sinh cosh cosh = cosh + sinh rcosh = ln{ + } ( ) rsinh = ln{ + + } + rtnh = ln ( < ) FP3 pst questions - conics Conics Ellipse Prol Hperol Rectngulr Hperol Stndrd
n f(x i ) x. i=1 In section 4.2, we defined the definite integral of f from x = a to x = b as n f(x i ) x; f(x) dx = lim i=1
The Fundmentl Theorem of Clculus As we continue to study the re problem, let s think bck to wht we know bout computing res of regions enclosed by curves. If we wnt to find the re of the region below the
Integration. antidifferentiation
9 Integrtion 9A Antidifferentition 9B Integrtion of e, sin ( ) nd os ( ) 9C Integrtion reognition 9D Approimting res enlosed funtions 9E The fundmentl theorem of integrl lulus 9F Signed res 9G Further
Definition of Continuity: The function f(x) is continuous at x = a if f(a) exists and lim
Mth 9 Course Summry/Study Guide Fll, 2005 [1] Limits Definition of Limit: We sy tht L is the limit of f(x) s x pproches if f(x) gets closer nd closer to L s x gets closer nd closer to. We write lim f(x)